Essential Geometry Definitions and Formulas

Fundamental Geometric Concepts

Basic Geometric Elements

• Line: A straight path without beginning or end, extending infinitely in both directions.
• Ray: A straight path that has a beginning point but no end, extending infinitely in one direction.
• Segment: A straight path delimited by two distinct endpoints.
• Angle: The opening formed by two rays that share a common starting point (vertex).

Relationships Between Lines

• Intersecting Lines (Secant): Lines that cross each other at a single point.
• Parallel Lines: Lines that lie in the same plane and never intersect, having nothing in common.
• Coincident Lines: Lines that occupy the exact same position, sharing all points in common.
• Perpendicular Lines: Lines that intersect to form four equal (90-degree) angles, dividing the plane into four equal parts.

Polygons: General Properties

• Polygon: A closed, flat figure bounded by three or more line segments (sides).
• Regular Polygon: A polygon where all sides are of equal length and all interior angles are of equal measure.
• Irregular Polygon: A polygon that is not regular; its sides or angles (or both) are not all equal.

Sum of Interior Angles of a Polygon

The sum of the interior angles of an n-sided polygon is given by the formula: 180° × (n - 2).

Types of Polygons by Number of Sides

• Triangle: 3 sides
• Quadrilateral: 4 sides
• Pentagon: 5 sides
• Hexagon: 6 sides
• Heptagon: 7 sides
• Octagon: 8 sides
• Nonagon: 9 sides
• Decagon: 10 sides
• Hendecagon: 11 sides
• Dodecagon: 12 sides

Triangles: Classification and Properties

Classification by Angles

• Acute Angle: An angle measuring less than 90°.
• Obtuse Angle: An angle measuring greater than 90° but less than 180°.
• Right Angle: An angle measuring exactly 90°.

Types of Triangles by Angles

• Right-angled Triangle: A triangle with one right angle.
• Acute-angled Triangle: A triangle with three acute angles.
• Obtuse-angled Triangle: A triangle with one obtuse angle.

Types of Triangles by Sides

• Equilateral Triangle: A triangle with three sides of equal length and three equal angles (each 60°).
• Isosceles Triangle: A triangle with two sides of equal length and two equal angles opposite those sides.
• Scalene Triangle: A triangle with all three sides of different lengths and all three angles of different measures.

Notable Lines and Points in a Triangle

• Median: A line segment connecting a vertex to the midpoint of the opposite side. The medians intersect at the barycenter (centroid).
• Perpendicular Bisector: A line perpendicular to a side of a triangle and passing through its midpoint. The perpendicular bisectors intersect at the circumcenter.
• Altitude: A line segment from a vertex perpendicular to the opposite side (or its extension). The altitudes intersect at the orthocenter.
• Angle Bisector: A line segment that divides an angle into two equal angles. The angle bisectors intersect at the incenter.

Pythagorean Theorem

In a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides (legs).
Formula: Hypotenuse² = Leg₁² + Leg₂² or c² = a² + b²

Quadrilaterals: Specific Types

• Parallelogram (Rhomboid): A quadrilateral with two pairs of parallel sides. Opposite sides are equal in length, and opposite angles are equal. It has no right angles unless it's a rectangle or square.
• Trapezoid (Trapezium): A quadrilateral with at least one pair of parallel sides (called bases) and two non-parallel sides (legs).
  • Right Trapezoid: A trapezoid with two right angles.
  • Isosceles Trapezoid: A trapezoid where the non-parallel sides (legs) are of equal length, and base angles are equal.
  • Scalene Trapezoid: A trapezoid where all four sides are of different lengths and no angles are equal.

Circles: Elements and Definitions

• Radius: A line segment connecting the center of the circle to any point on its circumference.
• Chord: A line segment joining any two points on the circumference of a circle.
• Diameter: A chord that passes through the center of the circle. It is the longest chord and is twice the length of the radius.
• Arc: A continuous part of the circumference of a circle between two points on it.

Geometric Formulas: Perimeter and Area

Perimeter Formulas

• Irregular Polygon: The perimeter is the sum of the lengths of all its sides.
  Formula: P = Side₁ + Side₂ + Side₃ + ... + Sideₙ
• Regular Polygon: The perimeter is the number of sides multiplied by the length of one side.
  Formula: P = Number of Sides × Side Length
• Circumference of a Circle: The distance around the circle.
  Formula: C = π × Diameter or C = 2 × π × Radius
• Arc Length: The length of a segment of the circumference.
  Formula: Arc Length = (Angle in degrees / 360°) × 2 × π × Radius

Area Formulas

• Rectangle: Area = Base × Height
• Square: Area = Side × Side = Side²
• Rhombus: Area = (Diagonal₁ × Diagonal₂) / 2
• Parallelogram (Rhomboid): Area = Base × Height
• Triangle: Area = (Base × Height) / 2
• Trapezoid: Area = ((Base₁ + Base₂) / 2) × Height
• Circle: Area = π × Radius²